Just to remember the basics of polar coordinates:
Polar Coordinates
Known that sec theta= 1/cos theta, the expression in polar coordinates may be so rewritten:
r=4*tan theta(1/cos theta)
Toward the convertion we know that:
r=sqrt(x^2+y^2)
theta=arc tan(y/x)
Then tan theta=tan (arc tan (y/x))=y/x
To complete the convertion we only need to determine cos theta in function of x and y:
tan theta = y/x => sin theta/cos theta =y/x => sqrt (1-cos^2 theta)/cos theta =y/x => (1-cos^2 theta)/cos^2 theta=y^2/x^2 => x^2-x^2*cos^2 theta = y^2*cos^2 theta => cos^2 theta*(x^2+y^2)=x^2 => cos theta =x/sqrt(x^2+y^2)
Substituting r, tan theta and cos theta, by the corresponding functions in x and y, the original expression becomes:
sqrt(x^2+y^2)=4.(y/x)(1/(x/sqrt(x^2+y^2))) => cancel(sqrt(x^2+y^2))*(x/cancel(sqrt(x^2+y^2)))=4.(y/x) => x^2=4y
Testing the result (or reconverting it to polar coordinates):
x^2=4y => (r*cos theta)^2=4*r*sin theta => r ^cancel(2)*cos^2 theta=4*cancel(r)*sin theta => r=4(sin theta/cos theta)(1/(cos theta)) [This expression is equivalent to the original expression. Thus the resulting expression (in x and y) is correct.]
